Election Forensics Toolkit

BasicElectionForensics()

The BasicElectionForensics function is a comprehensive R tool designed to perform statistical analysis on election data to detect potential irregularities or fraud patterns. This function implements multiple forensic methods commonly used in election integrity research and provides both statistical results and significance testing through bootstrap methods.

BasicElectionForensics(data, Candidates, Level="National", 
                      TotalReg, TotalVotes, Methods, R=1000, cores=2)

Input

Parameter	Type	Description
data	data.frame	The input dataset containing election results
Candidates	vector	Variable names referring to vote counts for candidates/parties
Level	string	Variable name depicting the level of analysis (default: “National”)
TotalReg	string	Variable name for the total number of eligible voters
TotalVotes	string	Variable name for the total number of ballots cast
Methods	vector	List of forensic methods to apply (see Methods section)
R	numeric	Number of bootstrap simulations (default: 1000)
cores	numeric	Number of cores for parallel computing (default: 2)

Output

Output	Type	Description
table	data.frame	Numerical results of the election forensic tests
tex	xtable, data.frame	LaTeX-formatted results table suitable for academic use
html	datatables, htmlwidget	Interactive HTML table with color-coded significance
sigMatrix	matrix, array	Binary significance matrix for further statistical analysis

Statistical Distribution Tests

• _2BL - Second-digit mean test (Benford’s Law analysis)
• LastC - Last-digit mean test for uniform distribution
• P05s - Percentage last-digit 0/5 indicator (for percentages)
• C05s - Count last-digit 0/5 indicator (for vote counts)
• Skew - Skewness measure (asymmetry from normal distribution)
• Kurt - Kurtosis measure (tail heaviness compared to normal distribution)
• DipT - Unimodality test (Hartigan’s dip test)
• Sobyanin - Sobyanin-Sukhovolsky measure (turnout-vote share relationship)
• Correlation - Correlation coefficient between turnout and vote share

Key Features

• Uses nonparametric bootstrap with configurable number of simulations (R parameter)
• Provides confidence intervals for statistical significance
• Supports parallel processing for improved performance
• Works with various election data formats
• Handles missing data appropriately
• Supports multi-level analysis (e.g., national, regional, local)

Usage Example

library(EFToolkit)

# Load election data
dat <- read.csv(system.file("extdata/Albania2013.csv", package="EFToolkit"))

# Run forensics analysis
results <- BasicElectionForensics(
  dat,
  Candidates = c("C035", "C050"),
  Level = "Prefectures", 
  TotalReg = "Registered",
  TotalVotes = "Ballots",
  Methods = c("P05s", "C05s", "_2BL", "Sobyanin", 
              "DipT", "Skew", "Kurt", "Correlation"),
  cores = 2, 
  R = 100  # Reduced for faster computation in example
)

# View results
print(results$table)

Basic Election Forensics Table

Interpretation Guidelines

Test	No fraud	Interpretation
Second-digit mean (2BL)	4.187	Values close to 4.19 are consistent with Benford’s Law. Systematic deviations (too low or too high) may indicate artificial rounding or human fabrication.
Last-digit mean (LastC)	4.5	Randomly distributed last digits should average 4.5. Substantial deviations suggest that numbers may not be uniformly random (e.g., preferences for certain digits).
Count of last-digit 0/5 ind. mean (C05s)	0.2	About 20% of values should end in 0 or 5. Excess frequency of 0s or 5s may reflect rounding or strategic reporting.
Perc. last-digit 0/5 ind. mean (P05s)	0.2	Same logic applies to percentages; deviations from 0.2 may signal manipulation of turnout or result percentages.
Skewness (Skew)	0	A symmetric distribution has skewness near zero. Positive skew indicates a longer right tail (many low but a few very high values); negative skew the opposite. Significant skewness may indicate anomalous clustering of results.
Kurtosis (Kurt)	3	A normal distribution has kurtosis of 3. Higher values (>3) indicate peakedness (results too concentrated), while lower values (<3) suggest excessive dispersion.
Unimodality test p-value (DipT)	>0.05	A p-value greater than 0.05 supports unimodality (single peak). Values below 0.05 indicate multimodality, possibly reflecting a mixture of normal and manipulated results.
Sobyanin–Sukhovolsky	near 0	Captures the degree of association between turnout and vote share. Under normal conditions, this relationship should be weak or nonexistent; strong positive association may indicate manipulated results.
Correlation coefficient (Corr)	near 0	Measures the correlation between turnout and vote share across precincts. Values close to zero are expected in competitive elections; high positive correlations suggest manipulated results.

BuildMap()

The BuildMap function is a specialized R visualization tool designed to create choropleth maps from election forensics analysis results. This function takes the output from BasicElectionForensics() and combines it with geographic data to produce spatial visualizations of statistical anomalies and patterns in election data.

BuildMap(eforensicsdata, geodata, Geoindex, Colorsig=FALSE, xlab="")

Input

Parameter	Type	Description
eforensicsdata	list	Output object from BasicElectionForensics() function
geodata	sf object	Spatial data frame containing geographic boundaries
Geoindex	string	Name of index variable from sf object used to merge geodata with forensics results
Colorsig	logical	If TRUE, only statistically significant estimates are mapped (default: FALSE)
xlab	string	X-axis label or subtitle for the maps

Output

Output	Type	Description
figures	list	Collection of generated plots (either spplot or tmap objects)
Colorsig	logical	Indicator of whether significance-based coloring was applied (TRUE/FALSE)
shpdata	sf (simple features)	Spatial dataset with election forensics results merged with geodata
creationdate	POSIXct	Timestamp of when the output was created

Key Features

Spatial Data Integration

• Seamlessly merges election forensics results with geographic boundaries
• Supports sf (Simple Features) spatial data format
• Handles missing data and geographic mismatches gracefully

Flexible Visualization Options

• Standard Mode: Displays all forensic values with continuous color scales
• Significance Mode: Highlights only statistically significant results
• Customizable color schemes using ColorBrewer palettes

Multi-Method Mapping

• Automatically generates maps for all forensic methods in the input data
• Creates separate visualizations for each candidate-method combination
• Supports multiple candidates and analysis levels simultaneously

Usage Example

library(EFToolkit)
library(sf)

# Load election data
dat <- read.csv(system.file("extdata/Albania2013.csv", package="EFToolkit"))

# Run forensics analysis
eldata <- BasicElectionForensics(dat,
                                Candidates=c("C035", "C050"),
                                Level="Prefectures", TotalReg="Registered",
                                TotalVotes="Ballots",
                                Methods=c("P05s", "C05s"), R=100)

# Load geographic data
geodata <- st_read(system.file("extdata/Albania2013_prefectures.shp",
                          package="EFToolkit"), quiet = TRUE)

# Create maps
figures <- BuildMap(eforensicsdata=eldata, geodata=geodata, Geoindex="Level")

# Access individual maps
print(figures$figures$Turn_P05s)  # Turnout P05s method map
print(figures$figures$C035_P05s)  # Candidate C035 P05s method map

The function generates choropleth maps showing spatial patterns of election forensics indicators:

Image 1	Image 2	Image 3	Image 4

Technical Implementation Details

Spatial Data Handling

• Uses sf package for modern spatial data processing
• Maintains coordinate reference systems throughout analysis
• Supports various geographic file formats (shapefile, GeoJSON, etc.)
• Implements ColorBrewer “OrRd” (Orange-Red) palette by default
• Provides good contrast for highlighting anomalies
• Color-blind friendly options available through ColorBrewer

Color Intensity Interpretation

• Light Colors: Values close to expected/normal ranges
• Medium Colors: Moderate deviations from expected patterns
  
• Dark Colors: Strong deviations suggesting potential irregularities
• No Color/White: Missing data or areas excluded from analysis

Spatial Pattern Analysis

• Clustering: Adjacent areas with similar colors may indicate systematic issues
• Isolated Anomalies: Single areas with extreme values warrant individual investigation
• Border Effects: Patterns along administrative boundaries may suggest institutional factors
• Urban/Rural Differences: Different patterns by area type are common

ClusterAnalysis()

The ClusterAnalysis function implements sophisticated spatial clustering tests to identify geographic patterns and anomalies in election forensics data. This function uses two complementary statistical methods - Getis-Ord Gi* and Local Moran’s I - to detect spatial clusters of unusual values, helping researchers identify areas where election irregularities may be spatially correlated.

ClusterAnalysis(geodata, Vars, IndexCL=NULL, cores=2)

Input

Parameter	Type	Description
geodata	sf object/list	Spatial data within a list (supports both polygon and point sf objects) or BuildMap object
Vars	character vector	Variable names used for geographic clustering tests
IndexCL	string	Index variable name used for merging polygon and point sf objects (optional)
cores	numeric	Number of cores for parallel computing (default: 2)

Output

Output	Type	Description
Moran's I for <Var>	ggplot	A ggplot map visualizing local Moran’s I cluster results for variable <Var>.
Getis-Ord for <Var>	ggplot	A ggplot map visualizing Getis-Ord G* hot- and cold-spots for variable <Var>.

Spatial Clustering Methods

Local Moran’s I

• Purpose: Identifies local clusters and spatial outliers
• Detects: Four types of spatial associations:
  • HH (High-High): High values surrounded by high values
  • LL (Low-Low): Low values surrounded by low values
    
  • HL (High-Low): High values surrounded by low values
  • LH (Low-High): Low values surrounded by high values

Getis-Ord Gi* Statistic

• Purpose: Identifies hot spots and cold spots
• Detects: Two types of spatial concentrations:
  • Hot Spots: Areas with significantly high values clustered together
  • Cold Spots: Areas with significantly low values clustered together

The function generates spatial cluster analysis maps showing different types of spatial patterns:

Image 1	Image 2	Image 3	Image 4

Spatial clustering analysis Spatial clustering analysis for Albanian election data - (1) Local Moran’s I for P05s method, (2) Getis-Ord hot/cold spots for P05s, (3) Local Moran’s I for C05s method, (4) Getis-Ord analysis for C05s method

library(EFToolkit)
library(sf)

# Load election data
dat <- read.csv(system.file("extdata/Albania2013.csv", package="EFToolkit"))

# Obtain election forensics estimates
eldata <- BasicElectionForensics(dat,
                                Candidates=c("C035", "C050"),
                                Level="Prefectures", TotalReg="Registered",
                                TotalVotes="Ballots",
                                Methods=c("P05s", "C05s"), R=100)

# Create the map with results
geodata <- st_read(system.file("extdata/Albania2013_prefectures.shp", 
                              package="EFToolkit"), quiet = TRUE)
figures <- BuildMap(eforensicsdata=eldata, geodata=geodata, 
                   Geoindex="Level", Colorsig=FALSE)

# Using the mapped results, implement cluster analysis
cluster_results <- ClusterAnalysis(figures, Vars=c("C050_P05s", "C050_C05s"))

# View specific cluster maps
print(cluster_results[["Moran's I for C050_P05s"]])
print(cluster_results[["Getis-Ord for C050_P05s"]])

Key Features

Advanced Spatial Statistics

• Permutation-based Testing: Uses Monte Carlo permutation tests for statistical significance
• False Discovery Rate Control: Implements Benjamini-Hochberg FDR correction for multiple testing
• Parallel Processing: Supports multi-core computation for large datasets

Flexible Input Handling

• Multiple Data Types: Works with polygon and point spatial data
• Integration Ready: Seamlessly processes BuildMap function outputs
• Missing Data Handling: Robust treatment of missing or invalid data points

Comprehensive Visualization

• Color-coded Results: Uses standardized color schemes for easy interpretation
• Multiple Significance Levels: Shows results at 90%, 95%, and 99% confidence levels
• Interactive Plotting: Generates ggplot2 objects for further customization

Color Coding System

Local Moran’s I Colors

Pattern	99% Level	95% Level	90% Level	Description
HH	Dark Red	Light Red	Pink	High-High clusters
LL	Dark Blue	Light Blue	Light Gray	Low-Low clusters
HL	Dark Orange	Light Orange	Yellow	High-Low outliers
LH	Dark Green	Light Green	Light Green	Low-High outliers
NS	Light Gray	Light Gray	Light Gray	Not significant

Getis-Ord Gi* Colors

Pattern	Description	Color
Hot Spot (99%)	Highly significant hot spot	Dark Red
Hot Spot (95%)	Significant hot spot	Medium Red
Hot Spot (90%)	Moderately significant hot spot	Light Red
Cold Spot (99%)	Highly significant cold spot	Dark Blue
Cold Spot (95%)	Significant cold spot	Medium Blue
Cold Spot (90%)	Moderately significant cold spot	Light Blue
Not Significant	No significant clustering	Light Gray

Local Moran’s I Results

• High-High Clusters (Red): Areas with high forensic values surrounded by similar high values
  • Interpretation: Potential systematic irregularities in neighboring areas
  • Action: Priority areas for detailed investigation
• Low-Low Clusters (Blue): Areas with low forensic values surrounded by similar low values
  • Interpretation: Regions with consistently normal patterns
  • Action: Lower priority for investigation
• High-Low Outliers (Orange): High values surrounded by low values
  • Interpretation: Isolated anomalies or data quality issues
  • Action: Investigate for administrative or data collection problems
• Low-High Outliers (Green): Low values surrounded by high values
  • Interpretation: Unusually clean areas within problematic regions
  • Action: Verify data accuracy or investigate protective factors

Getis-Ord Results

• Hot Spots (Red Shades): Statistically significant concentrations of high values
  • Interpretation: Geographic clustering of potential irregularities
  • Action: Focus investigative resources on these areas
• Cold Spots (Blue Shades): Statistically significant concentrations of low values
  • Interpretation: Areas with consistently normal electoral patterns
  • Action: Use as baseline comparisons or control areas

NonparamElectionForensics()

The NonparamElectionForensics function implements a revised version of Shpilkin’s method for detecting election fraud through nonparametric analysis of vote distributions. This method identifies anomalous voting patterns by analyzing the relationship between turnout and candidate support, detecting artificial vote inflation through statistical modeling of “clean” electoral behavior.

NonparamElectionForensics(data, Candidates, CandidatesText=NULL,
                         MainCandidate, TotalReg, TotalVotes=NULL,
                         Level=NULL, MaxThreshold=0.8,
                         FigureName, setcolors=NULL,
                         precinctLevel=TRUE, computeSD=NULL,
                         sims=10, mode_search=list(npeaks=5, sortstr=TRUE,
                                                  minpeakdistance=1, pick_by="height"),
                         man_turnout=NULL, grid_type="1D")

Input

Parameter	Type	Description
data	data.frame	Electoral data containing vote counts and registration information
Candidates	vector	Variable names for all candidates/parties in the election
CandidatesText	vector	Display names for candidates/parties (uses Candidates if NULL) (default: NULL)
MainCandidate	string	Variable name for main/incumbent candidate
TotalReg	string	Variable name for total number of eligible voters
TotalVotes	string	Variable name for total ballots cast (computed from candidates if NULL) (default: NULL)
Level	string	Variable for analysis level (default: “National”)
MaxThreshold	numeric	Anomalous turnout threshold (default: 0.8)
FigureName	string	Title for generated figures
setcolors	vector	Custom color palette (random if NULL) (default: NULL)
precinctLevel	logical	Whether to compute precinct-level estimates (default: TRUE)
computeSD	string	Standard error method: “parametric” or “nonparametric” (default: NULL)
sims	numeric	Number of simulations for uncertainty estimation (default: 10)
mode_search	list	Clean peak search parameters
man_turnout	numeric	Manual clean peak turnout override (default: NULL)
grid_type	string	Estimation approach: “1D” or “2D” (default: “1D”)

The mode_search parameter accepts a list with the following components:

Parameter	Description
npeaks	Maximum number of peaks to identify
sortstr	Whether to sort peaks by strength
minpeakdistance	Minimum distance between peaks
pick_by	Peak selection method: “area”, “height”, “cluster”, “quantile”, or “elipse”

Output

Output	Type	Description
list_graphs	list	Collection of generated plots (ggplot2/plotly objects)
base_stats	list	Basic fraud statistics for the whole dataset
sim_all_stats	list	Simulation statistics for the whole dataset (if computeSD specified)
sim_hetero_stats_base	data.frame	Base statistics for regional analyses (if Level != “National”)
sim_hetero_stats_sims	data.frame	Simulation statistics for regional analyses (if Level != “National”)
fraud_precinct_data	data.frame	Precinct-level fraud estimates with uncertainty measures
data	data.frame	Original input data with computed variables
Level	character	Analysis level used in the function
creationdate	POSIXct	Timestamp of when the output was created

Precinct-Level Fraud Data Structure

When precinctLevel=TRUE, the fraud_precinct_data component contains:

Column	Description
id	Unique precinct identifier
base.fraud.votes	Point estimate of fraudulent votes
sim.precinct_mean	Mean fraud estimate from simulations
sim.precinct_sd	Standard deviation of fraud estimates
sim.sig_all	Statistical significance indicator
precinct_mean_hetero	Regional-level fraud estimates (when Level != “National”)

Primary Outputs

• Official Turnout: Reported voter participation rate
• Real Turnout: Estimated legitimate turnout (clean peak)
• Official Support: Reported candidate vote share
• Real Support: Estimated legitimate support in clean regions
• Ballot Stuffing: Votes added through turnout inflation
• Ballot Switching: Votes transferred between candidates
• Total Fraud: Combined fraudulent votes
• Proportional Fraud: Fraud as percentage of total votes

Uncertainty Quantification

When computeSD is specified, the function provides:

• Parametric: Assumes binomial distributions for vote generation
• Nonparametric: Uses bootstrap resampling for robust estimates

Clean Peak Detection Methods

• “height”: Selects clean peak with highest vote count
• “area”: Chooses clean peak with largest area under curve
• “cluster”: Uses clustering to identify clean peak
• “quantile”: Employs mixture models on turnout distribution for clean peak detection
• “elipse”: Uses robust covariance estimation for clean peak detection

Grid Types

• 1D Grid: Uses 1D grid of turnout distribution to identify clean peak
• 2D Method: Uses 2D grid of joint distribution of turnout and incumbent’s vote share

Example 1: National-Level Analysis with 1D Grid

library(EFToolkit)

# Load Russian 2000 election data
dat <- read.csv("electionfraud2000.csv")

# National analysis using 1D estimation method
res1 <- NonparamElectionForensics(dat, 
                                 Candidates = paste("P", 1:12, sep=""),
                                 CandidatesText = c("Stanislav Govorukhin", "Umar Dzhabrailov",  
                                                    "Vladimir Zhirinovsky", "Gennady Zuganov", 
                                                    "Ella Pamfilova", "Alexei Podberezkin", 
                                                    "Vladimir Putin", "Yuri Skuratov",
                                                    "Konstantin Titov", "Aman Tuleev",
                                                    "Grigorii Yavlinsky", "Against All"),
                                 MainCandidate = "P7",
                                 TotalReg = "NVoters",
                                 TotalVotes = "NValid",
                                 Level = "National",
                                 MaxThreshold = 0.8,
                                 mode_search = list(npeaks = 5, sortstr = TRUE,
                                                    minpeakdistance = 1, pick_by = "height"),
                                 FigureName = "Russian Presidential Elections, 2000",
                                 setcolors = c("royalblue2", "springgreen1","blue", 
                                               "red", "green","brown2",
                                               "darkgreen", "yellow", "lawngreen", 
                                               "purple","chartreuse1", "orange"),
                                 precinctLevel = TRUE, 
                                 computeSD = "nonparametric",
                                 sims = 10, 
                                 grid_type = "1D")

# Summary of precinct-level fraud estimates
total_fraud <- sum(res1$fraud_precinct_data$base.fraud.votes, na.rm=TRUE)
# Result: 2,685,246 fraudulent votes detected

# Statistically significant fraud only
significant_fraud <- sum(res1$fraud_precinct_data$sim.precinct_mean[
  res1$fraud_precinct_data$sim.sig_all==TRUE], na.rm=TRUE)
# Result: 811,501 significant fraudulent votes

> res1$base_stats
$`Whole dataset`
official_turnout     real_turnout official_support     real_support  ballot_stuffing ballot_switching 
    6.820000e+01     6.700000e+01     5.330000e+01     5.200000e+01     1.258868e+06     1.426378e+06 
     total_fraud       prop_fraud 
    2.685246e+06     6.990511e-02 
    
# Display the table of region-level measures
View(round(res1$sim_hetero_stats_base, 3))

Image 1	Image 2	Image 3

Precinct-level results

Example 2: Regional-Level Analysis with 1D Grid

# Regional analysis across all federal subjects
res2 <- NonparamElectionForensics(dat, 
                                 Candidates = paste("P", 1:12, sep=""),
                                 CandidatesText = c("Stanislav Govorukhin", "Umar Dzhabrailov",  
                                                    "Vladimir Zhirinovsky", "Gennady Zuganov", 
                                                    "Ella Pamfilova", "Alexei Podberezkin", 
                                                    "Vladimir Putin", "Yuri Skuratov",
                                                    "Konstantin Titov", "Aman Tuleev",
                                                    "Grigorii Yavlinsky", "Against All"),
                                 MainCandidate = "P7",
                                 TotalReg = "NVoters",
                                 TotalVotes = "NValid",
                                 Level = "regname",  # Regional analysis
                                 MaxThreshold = 0.8,
                                 mode_search = list(npeaks = 5, sortstr = TRUE,
                                                    minpeakdistance = 1, pick_by = "height"),
                                 FigureName = "Russian Presidential Elections, 2000",
                                 setcolors = c("royalblue2", "springgreen1","blue", 
                                               "red", "green","brown2",
                                               "darkgreen", "yellow", "lawngreen", 
                                               "purple","chartreuse1", "orange"),
                                 precinctLevel = TRUE, 
                                 computeSD = "nonparametric",
                                 sims = 10, 
                                 grid_type = "1D")

# Regional fraud estimates
regional_fraud <- sum(res2$fraud_precinct_data$precinct_mean_hetero, na.rm=TRUE)
# Result: 1,693,742 fraudulent votes across regions

# Statistically significant regional fraud
significant_regional_fraud <- sum(res2$fraud_precinct_data$precinct_mean_hetero[
  res2$fraud_precinct_data$sim.sig_all==TRUE], na.rm=TRUE)
# Result: 1,125,151 significant fraudulent votes

Image 1	Image 2	Image 3	Image 4

Region-level results

Example 3: National Analysis with 2D Grid

# 2D analysis using parametric uncertainty estimation
res3 <- NonparamElectionForensics(dat, 
                                 Candidates = paste("P", 1:12, sep=""),
                                 CandidatesText = c("Stanislav Govorukhin", "Umar Dzhabrailov",  
                                                    "Vladimir Zhirinovsky", "Gennady Zuganov", 
                                                    "Ella Pamfilova", "Alexei Podberezkin", 
                                                    "Vladimir Putin", "Yuri Skuratov",
                                                    "Konstantin Titov", "Aman Tuleev",
                                                    "Grigorii Yavlinsky", "Against All"),
                                 MainCandidate = "P7",
                                 TotalReg = "NVoters",
                                 TotalVotes = "NValid",
                                 Level = "National",
                                 MaxThreshold = 0.8,
                                 mode_search = list(npeaks = 5, sortstr = TRUE,
                                                    minpeakdistance = 1, pick_by = "height"),
                                 FigureName = "Russian Presidential Elections, 2000",
                                 setcolors = c("royalblue2", "springgreen1","blue", 
                                               "red", "green","brown2",
                                               "darkgreen", "yellow", "lawngreen", 
                                               "purple","chartreuse1", "orange"),
                                 precinctLevel = TRUE, 
                                 computeSD = "parametric",
                                 sims = 10, 
                                 grid_type = "2D")

# 2D method fraud estimates
fraud_2d <- sum(res3$fraud_precinct_data$sim.precinct_mean, na.rm=TRUE)
# Result: -15,724,742 (negative values suggest model limitations)

# Significant fraud in 2D analysis
significant_2d_fraud <- sum(res3$fraud_precinct_data$sim.precinct_mean[
  res3$fraud_precinct_data$sim.sig_all==TRUE], na.rm=TRUE)
# Result: 0 (no significant fraud detected with 2D method)

# Display the table of region-level measures
View(round(res3$sim_hetero_stats_base, 3))

Image 1	Image 2	Image 3

Example 4: Analysis of Selected Regions with 2D Grid

# Focus on specific regions of interest
selected_regions <- c("Respublika Dagestan", "Gorod Moskva", 
                      "Samarskaya Oblast`", "Volgogradskaya Oblast`")  
dat_subset <- dat[dat$regname %in% selected_regions,]

res4 <- NonparamElectionForensics(dat_subset, 
                                 Candidates = paste("P", 1:12, sep=""),
                                 CandidatesText = c("Stanislav Govorukhin", "Umar Dzhabrailov",  
                                                    "Vladimir Zhirinovsky", "Gennady Zuganov", 
                                                    "Ella Pamfilova", "Alexei Podberezkin", 
                                                    "Vladimir Putin", "Yuri Skuratov",
                                                    "Konstantin Titov", "Aman Tuleev",
                                                    "Grigorii Yavlinsky", "Against All"),
                                 MainCandidate = "P7",
                                 TotalReg = "NVoters",
                                 TotalVotes = "NValid",
                                 Level = "regname",
                                 MaxThreshold = 0.8,
                                 mode_search = list(npeaks = 5, sortstr = TRUE,
                                                    minpeakdistance = 1, pick_by = "height"),
                                 FigureName = "Russian Presidential Elections, 2000",
                                 setcolors = c("royalblue2", "springgreen1","blue", 
                                               "red", "green","brown2",
                                               "darkgreen", "yellow", "lawngreen", 
                                               "purple","chartreuse1", "orange"),
                                 precinctLevel = TRUE, 
                                 computeSD = "nonparametric",
                                 sims = 10, 
                                 grid_type = "2D")

# Access regional comparison results
print(res4$stats_summary)

Image 1	Image 2	Image 3

Image 4	Image 5	Image 6

Precinct-level results

Advanced Features

Multi-Level Analysis

When Level parameter specifies administrative units, the function: - Performs analysis for the entire dataset - Conducts separate analyses for each administrative unit - Aggregates results across regions - Provides comparative statistics

Robust Peak Detection

The algorithm implements multiple fallback strategies: 1. Primary method specified in pick_by 2. Alternative clustering approaches if primary fails 3. Simple peak detection as ultimate fallback 4. Manual override through man_turnout parameter

Precinct-Level Estimation

When precinctLevel=TRUE, the function: - Estimates fraud at individual precinct level - Uses post-stratification for statistical adjustment - Provides significance testing for precinct estimates - Enables spatial analysis integration

Best Practices

Method Selection

1. Start with 1D analysis for initial exploration
2. Use regional-level analysis for heterogeneous countries
3. Apply nonparametric uncertainty for robust estimates
4. Test multiple pick_by methods for sensitivity analysis

Parameter Tuning

1. Increase sims for more precise uncertainty estimates
2. Adjust MaxThreshold based on country-specific context
3. Experiment with mode_search parameters for optimal peak detection
4. Use custom setcolors for publication-quality visualizations

Result Validation

1. Compare across estimation methods (1D vs 2D)
2. Examine statistical significance alongside magnitude
3. Cross-validate with other forensic indicators
4. Consider substantive electoral context

Finite Mixture Model()

ComputeFiniteMixtureModel - A legacy implementation of Walter Mebane’s Finite Mixture Model for electoral data analysis.

The model uses Bayesian estimation techniques with EM-algorithm-like iterations to estimate the posterior probabilities of each precinct belonging to each fraud category.

⚠️ Important Note: This function is legacy code that is no longer actively maintained or supported. It may have dependencies on outdated packages or contain unoptimized algorithms.

ComputeFiniteMixtureModel(dat, MainCandidate = "Votes", TotalReg = "NVoters", 
                         TotalVotes = "NValid", cores = 2, itstartmax = 1)

Input

Parameter	Type	Description
dat	data.frame	Electoral dataset containing voting data
MainCandidate	character	Variable name for the major/incumbent candidate votes (default: “Votes”)
TotalReg	character	Variable name for total number of eligible voters (default: “NVoters”)
TotalVotes	character	Variable name for total number of ballots cast (default: “NValid”)
cores	integer	Number of cores for parallel computing (default:2)
itstartmax	integer	Maximum number of iterations for optimization (default: 1)

Output

Returns a list containing FMM estimates with the following structure:

list(
  FF_null = matrix,    # Null model results (estimates and standard deviations)
  FFlist_null = list,  # Full null model output including posterior probabilities
  FF = matrix,         # Main model results (estimates and standard deviations)
  FFlist = list        # Full main model output including posterior probabilities
)

The output matrices contain the following parameters:

Parameter	Description
incremental	Proportion of incremental fraud
extreme	Proportion of extreme fraud
alpha	Fraud intensity parameter
turnout	Turnout rate parameter
winprop	Winning proportion parameter
sigma	Standard deviation for vote proportions
stdAtt	Standard deviation for attendance
theta	Convergence test parameter
loglik	Log-likelihood value
df	Degrees of freedom

Usage Example

library(EFToolkit)

# Load sample data
dat <- read.csv(system.file("extdata/ruspres2020.csv", package = "EFToolkit"))
dat <- subset(dat, select = c("region", "NVoters", "NValid", "Votes"))
datc <- dat[dat$region == "Volgogradskaya Oblast`", ]

# Run FMM analysis (commented out due to long computation time)
# res <- ComputeFiniteMixtureModel(datc,
#                                 MainCandidate = "Votes",
#                                 TotalReg = "NVoters", 
#                                 TotalVotes = "NValid")

Key Features

• Mixture Modeling: Implements a finite mixture model with multiple fraud components
• Parallel Computing: Supports multi-core processing for faster computation
• Robust Estimation: Uses genetic algorithms (rgenoud) for parameter optimization
• Statistical Inference: Provides estimates with standard errors

Limitations & Considerations

1. Computational Intensity: The function can be very slow for large datasets
2. Legacy Status: No longer actively supported or maintained
3. Algorithm Complexity: Implements sophisticated statistical models that may require domain expertise to interpret
4. Parameter Sensitivity: Results may be sensitive to starting values and iteration limits

Klimek Model()

ComputeKlimekModel - Implements the Klimek et al. (2012) simulation-based method for detecting electoral anomalies through histogram analysis of vote distributions.

⚠️ Legacy Function Notice: This function is legacy code that is no longer actively maintained or supported. It remains available for historical reference and research reproducibility but may have dependencies on outdated packages or contain unoptimized algorithms.

Key Features:

• Histogram-based analysis: Compares observed vote distributions with simulated ones
• Fraud simulation: Models incremental and extreme fraud scenarios
• Parameter estimation: Estimates fraud parameters through iterative simulation
• Statistical testing: Uses chi-square goodness-of-fit tests to evaluate model fit

ComputeKlimekModel(data, Candidates, Level = "National", TotalReg, TotalVotes, 
                   R = 1000, cores = 2)

Input

Parameter	Description
data	Data frame containing electoral data
Candidates	Variable name(s) for vote counts of candidates/parties
Level	Variable indicating geographic level of analysis (default: “National”)
TotalReg	Variable name for total number of eligible voters
TotalVotes	Variable name for total number of ballots cast
R	Number of simulations to run (default: 1000)
cores	Number of CPU cores for parallel computation (default: 2)

Output

Column	Description
Level	Geographic level of analysis
Candidate	Candidate/party name
KSimI	Estimated proportion of incremental fraud
KSimE	Estimated proportion of extreme fraud
KSimalpha	Fraud intensity parameter
KSimturnout	Estimated turnout rate parameter
KSimwinprop	Estimated winning proportion parameter
KSimsigma	Standard deviation for vote proportions
KSimstdAtt	Standard deviation for attendance rates
KSimtheta	Convergence test parameter
Obs	Number of observations used in analysis

Usage Example

library(EFToolkit)

# Load sample data
dat <- read.csv(system.file("Albania2013.csv", package = "EFToolkit"))

# Run Klimek analysis with reduced simulations for speed
klimek <- ComputeKlimekModel(dat, 
                            Candidates = "C050", 
                            Level = "National",
                            TotalReg = "Registered", 
                            TotalVotes = "Ballots", 
                            cores = 1, 
                            R = 100)

# Access results
print(klimek$table)      # Data frame with results
print(klimek$html)       # HTML formatted table
print(klimek$tex)        # LaTeX formatted table

The function computes several goodness-of-fit measures:

• Winner.HFit.Klimek: Histogram fit statistic for winning candidate
• Winner.HFit.chi2: Chi-square histogram fit
• Winner.Fit.chi2: Chi-square vote count fit
• Overall chi2: Comprehensive model fit statistic

Key Functions:

• Estimate(): Initial parameter estimation from vote distributions
• Sim_Vote(): Simulates electoral outcomes under fraud scenarios
• Sim.Histo(): Generates histogram distributions from simulated data
• Iteration_sim(): Main optimization loop for parameter estimation

Simulation Parameters:

• f1range: Incremental fraud proportion range (0.0 to 1.0)
• f2range: Extreme fraud proportion range (0.0 to 0.3)
  
• arange: Fraud intensity range (0.5 to 1.0)
• iterations: Number of optimization iterations (default: 10)

Interpretation Guidelines

• Low fraud estimates (KSimI, KSimE near 0): Suggest clean election
• High incremental fraud: Indicates widespread small-scale manipulation
• High extreme fraud: Suggests concentrated large-scale fraud
• Model fit statistics: Lower values indicate better model fit to observed data